Given a list L = {a, b1,…, bn} of positive integers with a= sumi bi, and a field (or ring of integers) kk, the script creates a polynomial ring S over kk with a×b1×…×bn variables, and a generic map
f: A →B1⊗…⊗Bn
of LabeledModules over S, where A is a free LabeledModule of rank a and Bi is a free LabeledModule of rank bi. We think of f as representing a tensor of type (a,b1,…,bn) made from the elementary symmetric functions.i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing |
i2 : f = flattenedGenericTensor({5,2,1,2},kk) o2 = | x_(0,0,0,0) x_(1,0,0,0) x_(2,0,0,0) x_(3,0,0,0) x_(4,0,0,0) | | x_(0,0,0,1) x_(1,0,0,1) x_(2,0,0,1) x_(3,0,0,1) x_(4,0,0,1) | | x_(0,1,0,0) x_(1,1,0,0) x_(2,1,0,0) x_(3,1,0,0) x_(4,1,0,0) | | x_(0,1,0,1) x_(1,1,0,1) x_(2,1,0,1) x_(3,1,0,1) x_(4,1,0,1) | 4 5 o2 : Matrix (kk[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ]) <--- (kk[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ]) 0,0,0,0 0,0,0,1 0,1,0,0 0,1,0,1 1,0,0,0 1,0,0,1 1,1,0,0 1,1,0,1 2,0,0,0 2,0,0,1 2,1,0,0 2,1,0,1 3,0,0,0 3,0,0,1 3,1,0,0 3,1,0,1 4,0,0,0 4,0,0,1 4,1,0,0 4,1,0,1 0,0,0,0 0,0,0,1 0,1,0,0 0,1,0,1 1,0,0,0 1,0,0,1 1,1,0,0 1,1,0,1 2,0,0,0 2,0,0,1 2,1,0,0 2,1,0,1 3,0,0,0 3,0,0,1 3,1,0,0 3,1,0,1 4,0,0,0 4,0,0,1 4,1,0,0 4,1,0,1 |
i3 : numgens ring f o3 = 20 |
i4 : betti matrix f 0 1 o4 = total: 4 5 -1: . 5 0: 4 . o4 : BettiTally |
i5 : S = ring f o5 = S o5 : PolynomialRing |
i6 : tensorComplex1 f o6 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0) 0 -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0) 0 -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0) 0 -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0) 0 -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0) 0 -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0) 0 -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0) 0 -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0) 0 -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0) 0 -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0) 0 | | -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0) -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0) -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0) -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0) -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0) -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0) -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0) -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0) -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0) -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0) | | -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1) -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1) -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1) -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1) -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) | | 0 -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1) 0 -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1) 0 -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1) 0 -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1) 0 -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1) 0 -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1) 0 -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1) 0 -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1) 0 -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1) 0 -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1) | 4 20 o6 : Matrix S <--- S |